Diffraction is an experimental technique in which radiation with a narrow range of wavelengths is shone on a sample. The radiation interacts with the electrons and/or nuclei in the sample and is scattered elastically. Interference within the scattered radiation creates an observable pattern that is characteristic of the molecular-scale structure of the sample. To be effective, the radiation should have a wavelength that is similar to the atomic scale. For example, X-rays, high energy electrons, and thermal neutrons may be used. X-rays are readily produced using laboratory sources and are non-destructive at sufficiently low doses. Therefore, X-ray diffraction (XRD) is the most commonly used diffraction technique.
Single crystal diffraction is used to determine the molecular scale structure of a crystalline form of matter using a single crystal. Diffracted radiation is measured as a function of scattering angle and relative orientation of the crystal. Typically thousands of coherent peaks in the interference pattern are used to determine the size and shape of the crystal unit cell and the positions of each atom in the crystal. While very useful, the need for a large crystal with few defects limits the application of single crystal diffraction.
If only a powder sample (as opposed to a single crystal) is available, then powder diffraction may be used to obtain a sub-set of the information available using single crystal diffraction. For a sample composed of many tiny and randomly oriented particles, the resulting diffraction pattern has continuous rings instead of discrete points and the diffraction pattern is largely independent of the orientation of the sample. Due to overlap of the rings, typically dozens of coherent peaks in the interference pattern are measured as a function of scattering angle. If X-rays are used for the radiation and the sample is a powder, then the technique is called X-ray powder diffraction (XRPD). FIG. 1 provides a schematic view of XRPD. As noted above, alternate forms of radiation may also be used.
Although the invention set forth herein will be described primarily with respect to XRPD, it should be understood that other forms of powder diffraction, such as electron and neutron powder diffraction, can also be used.
The term “crystalline” as used herein includes polycrystalline, microcrystalline, nanocrystalline, and partially or wholly crystalline substances, as well as disordered crystalline substances. Crystalline solid forms can include, for example, cocrystals, solvates and hydrates. Crystalline solid forms can also include polymorphs, which are different crystalline solid forms having the same chemical composition. Crystalline solid forms can include crystalline forms of salts of compounds, for instance, salts of pharmaceutical compounds. In addition, as used herein reference to a crystalline solid form of “a compound” includes a crystalline solid form comprising a compound and optionally one or more additional compounds or components, i.e., a multi-component system. For instance, a crystalline solid form of a compound includes a cocrystal and includes a salt of a compound.
The result of XRPD is an XRPD pattern. For samples composed of crystalline powders, the XRPD pattern may be a combination of sharp peaks and broad features. Sharp peaks are due to ordered crystalline regions in the sample. The sharp peaks occur at particular scattering angles (2θ) relative to the transmitted X-ray beam. The broad features may be due to a variety of factors, such as disorder in the sample, defects in the crystal, and/or scattering by air.
The XRPD pattern is a measured intensity (I) as a function of scattering angle (2θ). The positions of the peaks in the pattern and the relative intensities of the peaks are characteristic of a well-prepared sample of a particular crystalline material. Therefore, an XRPD pattern identifies a particular material, just as a fingerprint identifies a particular person. For many purposes, the XRPD pattern can be used directly without additional analysis.
Powder diffraction data may also be used to determine the crystallographic unit cell of the crystalline structure. Many methods of powder diffraction indexing are known. XRPD indexing, and in fact indexing of all powder diffraction data, is the process of determining the size and shape of the crystallographic unit cell consistent with the peak positions in a given XRPD pattern. Indexing does not make use of the relative intensity information in the XRPD pattern. The goal of the indexing process is the determination of three unit cell lengths (a,b,c), three unit cell angles (α,β,γ), and three Miller index labels (h,k,l) for each peak. The lengths are typically reported in Angstrom units (Å), and the angles in degree units. The Miller index labels are unitless integers. Successful indexing indicates that the sample is composed of one crystalline phase and is therefore not a mixture of crystalline phases. The indexing solution also provides a concise means to convey the positions of allowed peaks in an XRPD pattern. Other exemplary methods of powder diffraction indexing include, by way of example, Dicvol and X-cell, both of which are known to those of skill in the art.
Crystallographic unit cells are not unique. For any crystalline material, there is an infinite set of unit cells that may be used to describe the crystal structure of the material. There is, however, a unique unit cell called the reduced basis that has a minimal volume (V) and whose parameters (a,b,c,α,β,γ) conform to a set of rules. The reduced basis provides a systematic means for categorizing and comparing crystal forms. Reduced bases conform to one of 44 tabulated “characters.” Characters are a means of classification of reduced unit cells. Lattices belong to the same character if their reduced cells can be continuously deformed into one-another without changing the Bravais type and with continuous changes of the reduced lattice parameters. For each character there is a tabulated matrix transformation that generates a conventional unit cell from the reduced basis. The conventional cell parameters are most often reported as the outcome of XRPD indexing.
The invention makes use of an algorithm developed by the inventor, herein termed the “Triads Algorithm.” By applying the Triads Algorithm to measured instrumental data, the benefits of the invention, as described below, may be achieved.
In the Triads Algorithm described below, the overall strategy is to identify a member of the infinite set of unit cells that describe the lattice, then use well-established methods to transform the cell to its reduced basis and finally to the conventional unit cell.
The Bragg Equation (1) relates the Bragg angle (θ) to the order of the reflection (n), the radiation wavelength (λ), and the distance between Miller planes (dhkl):nλ=2dhkl sin(θ)  (1)Rearranging equation (1) provides an equation for the scattering vector magnitude (κhkl):
                              κ          hkl                =                                            2              ⁢                                                          ⁢              sin              ⁢                                                          ⁢                              (                θ                )                                      λ                    =                      n                          d              hkl                                                          (        2        )            Although the symbol k is often used for the magnitude of the scattering vector, here the symbol κ is used to avoid confusion with the Miller indices (h,k,l). The distance between Miller planes (dhkl) is a function of the Miller indices (h,k,l) and the crystallographic unit cell parameters (a,b,c,α,β,γ). This function is most conveniently expressed using matrix multiplication:
                                          (                          κ              hkl                        )                    2                =                              [                          h              ⁢                                                          ⁢              k              ⁢                                                          ⁢              l                        ]                    ·                      B                          _              _                                ·                      [                                                            h                                                                              k                                                                              l                                                      ]                                              (        3        )            where B is the Bragg matrix:
                              B                      _            _                          =                  [                                                                                          (                                          a                      *                                        )                                    2                                                                                                  (                                          a                      *                                        )                                    ⁢                                      (                                          b                      *                                        )                                    ⁢                                      cos                    ⁡                                          (                                              γ                        *                                            )                                                                                                                                        (                                          a                      *                                        )                                    ⁢                                      (                                          c                      *                                        )                                    ⁢                                      cos                    ⁡                                          (                                              β                        *                                            )                                                                                                                                                                (                                          a                      *                                        )                                    ⁢                                      (                                          b                      *                                        )                                    ⁢                                      cos                    ⁡                                          (                                              γ                        *                                            )                                                                                                                                        (                                          b                      *                                        )                                    2                                                                                                  (                                          b                      *                                        )                                    ⁢                                      (                                          c                      *                                        )                                    ⁢                                      cos                    ⁡                                          (                                              α                        *                                            )                                                                                                                                                                (                                          a                      *                                        )                                    ⁢                                      (                                          c                      *                                        )                                    ⁢                                      cos                    ⁡                                          (                                              β                        *                                            )                                                                                                                                        (                                          b                      *                                        )                                    ⁢                                      (                                          c                      *                                        )                                    ⁢                                      cos                    ⁡                                          (                                              α                        *                                            )                                                                                                                                        (                                          c                      *                                        )                                    2                                                              ]                                    (        4        )            and (a*,b*,c*,α*,β*,γ*) are reciprocal cell parameters. Equations (3) and (4) constitute the quadratic form of the Bragg Equation. It gives the peak position for a given Miller index triplet (h,k,l) given a unit cell of specified reciprocal cell parameters.
Simplified versions of the quadratic form are available for symmetric crystal systems. These simplifications are special cases of the general triclinic case provided in equations (3) and (4). Since the triclinic case is general, it is used in the implementation of the Triads Algorithm described below. Higher symmetry unit cells are recognized during the course of the algorithm.
Friedel's Law states that peaks labeled with Miller indices (h,k,l) and (−h,−k,−l) are indistinguishable in the absence of absorption effects. Slight deviations from Friedel's Law are used in the context of single crystal diffraction to determine absolute configuration of chiral molecules, but this is not relevant to XRPD since the opposing pairs of peaks overlap each other in an XRPD pattern. Since Friedel pairs always overlap in XRPD patterns, it is convenient to choose a naming convention that eliminates this ambiguity. In a preferred embodiment of the present invention, all Miller indices may be chosen such that the first non-zero index is positive. This embodiment is used in the examples below. In a further embodiment, all Miller indices may be chosen such that the first non-zero index is negative. In yet a further embodiment, a combination of positive and negative first non-zero Miller indices may be chosen.
Each of the infinite set of unit cells that describe a particular crystalline material yields reflections at the same set of peak positions (2θ) through the Bragg Equation. Larger unit cells will indicate additional peaks that are not indicated for a minimal volume unit cell, however. Since different unit cells have different unit cell parameters, the Miller indices labeling each peak are also different, but the set of distances between Miller planes is fixed. An example illustrating the equivalence of different unit cells is given in TABLE 1.
TABLE 1Exemplary comparison of Miller index labels for three different unitcells describing the same crystal with the same peak positions.Arbitrarily Specified Cella = 10.000 Å, b = 17.321 Å,Reduced BasisConventional Cellc = 26.458 Å, α = 10.89°a = b = c = 10.000 Å,a = b = c = 14.142 Å,Observed Peakβ = 55.46° γ = 54.74°α = β = γ = 60°α = β = γ = 90°Positions (2θ)(Triclinic)(Rhombohedral)(Face-Centered Cubic)10.827°(100), (123), (011), (012)(001), (010), (100), (111)(111)12.508°(112), (023), (111)(011), (101), (110)(002), (020), (200)17.724°(1-1-1), (223), (135),(01-1), (10-1), (1-10),(022), (202), (220)(134), (1-1-2), (001)(112), (121), (211)20.815°(212), (034), (1-2-3),(11-1), (1-11), (1-1-1),(113), (131), (311)(2, 3, 5), (101), (146),(021), (201), (210),(035), (124), (234),(012), (102), (120),(211), (122), (10-1)(122), (212), (221)21.752°(200), (246), (022), (024)(002), (020), (200), (222)(222)25.168°(224), (046), (222)(022), (202), (200)(004), (040), (400)For each of three different unit cells, Miller indices are given for each of the observed peaks. In TABLE 1 there are multiple Miller indices corresponding to each peak. This is a result of multiple lattice planes that diffract at the same Bragg angle and therefore overlap in the XRPD pattern. Such coincidence of reflections is common for symmetric unit cells such as the one used in TABLE 1. Each column in TABLE 1 is a different description of the same unit cell, but with different unit cell parameters and Miller index labels for the observed peaks. This demonstrates that there are multiple indexing solutions (column 1, for instance), any one of which may be reduced to the reduced basis (column 2) and then transformed to the conventional cell (column 3). This allows one to assign Miller indices in a convenient fashion, for example during the generation of trial solutions during the Triads Algorithm. Once the Algorithm generates a unit cell that is consistent with the observed peak list, then the cell can be reduced and transformed to the conventional cell.